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References
For Lp-convergence
L.L. HELMS: "Mean convergence of martingales". TAMS 87 (1985). p. 439–446.
For a.e. convergence and differentiation results
Y.S. CHOW: "Martingales in a σ-finite measure space, indexed by directed sets. TAMS 97 (1960), p. 254–285.
Y.S. CHOW: "Convergence theorems of martingales". ZW 1 (1962), p. 340–346.
K. KRICKEBERG: "Convergence of martingales with a directed index set". TAMS 83 (1956), p. 313–337.
K. KRICKEBERG and C. PAUC: "Martingales et dérivation". Bull. Soc. Math 91 (1963), p. 455–544.
For some information on the necessity of the Vitali conditions for a.e. convergence
MILLET and SUCHESTON: A characterization of Vitali conditions in terms of maximal inequalities, Ann. Prob. 8 (1980), p. 339–349.
For classical differentiation theory and Vitali's theorem
S. SAKS: "Theory of the Integral". Hafner, New-York, 1937.
References For the maximal and convergence theorems
CAIROLI: "Une inégalité pour martingales à indices multiples et ses applications", Séminaire de Strasbourg IV, p. 1–27.
For strong martingales and stopping lines
CAIROLI and WALSH: Stochastic integrals in the plane, Acta Math 134 (1975), p. 111–183.
WALSH: Convergence and regularity of strong matringales, Z W 46 (1979), p. 177–192.
WONG and ZAKAI: Weak martingales and stochastic integrals in the plane, Ann. Prob. 4 (1976), p. 570–586.
For differentiation
JESSEN, MARCINKIEWICZ and ZYGMUND: Note on the differentiability of multiple integrals, Fund. Math. 25, (1935), p. 217–234.
SAKS: Remark on the differentiability of the Lebesgue indefinite integral, Fund. Math. 22, (1934), p. 257–261.
ZYGMUND: On the differentiability of multiple integrals, Fund. Math. 23 (1934), p. 143–149.
SAKS: Theory of the Integral.
For biharmonic functions
CAIROLI: Ibid.
CAIROLI: Une représentation intégrale pour fonctions séparément excessives, Ann. Inst. Fourier 18, 1968, p. 317–338.
WALSH: Probability and a Dirichlet problem for multiply harmonic functions, Ann. Inst. Fourier 18, (1968), p. 221–229.
BROSSARD and CHEVALIER: Calcul stochactique etinégalités de normes pour les martingales bi-brownienne, Ann. Inst. Fourier 30, (1980).
For the law of large numbers of arrays
SMYTHE: Strong law of large numbers for r-dimensional arrays of random variables, Ann. Prob. 1, (1973), p. 164–170.
SMYTHE: Sums of independent random variables on partially ordered sets, Ann. Prob. 2 (1974), p. 906–917.
References Of historical interest
T. KITAGAWA, Analysis of variance applied to function spaces, Mem. Fac. Sci, Kyusu U. 6(1951), p. 41–53.
For general information
OREY, S. and PRUITT, W., Sample functions of the N-parameter Wiener process. Ann. Prob. 1, (1973), p. 138–163.
PYKE, R., Partial sums of matrix arrays and Brownian sheets. Stochastic Analysis, Kendall and Harding (ed.) Wiley, 1973.
These contain many references to the earlier literature.
For inequalities
GARSIA, Continuity properties of Gaussian processes with multidimensional time parameter. Proc. 6th Berkeley Symposium V.II (1971), p. 369–374.
For the vibrating string
CABANA, E. M., On barrier problems for the vibrating string, ZW 22 (1972), p. 13–24.
For level curves
KENDALL, W., Contours of Brownian processes with several-dimensional time, ZW 52 (1980) p. 269–276.
For singularties
OREY, S. and TAYLOR, S. J., How often on a Brownian path does the law of the iterated logarithm fail? Proc. London. Math. Soc. 28 (1974), p. 174–192.
WALSH, J. B., Propagation of singularities in the Brownian sheet, Ann. Prob. 10 (1982) p. 279–288.
For the law of large numbers
SMYTHE, R. T., Strong laws of large numbers for r-dimensional arrays of random variables, Ann. Prob. 1 (1973), p. 164–170.
SMYTHE, R. T., Sums of independent random variables on partially ordered sets, Ann. Prob. 2 (1974), p. 906–917.
There are numerous aspects of the Brownian sheet which we have not included here. For example, for a "strong Lévy's Markov property" see
EVSTIGNEEV, I. V., Markov times for random fields, Theor. Prob. Appl. 22 (1978), p. 563–569.
For local Times
MÃœLLER, D. W., Continuity of local time of Brownian motion with two-dimensional time, (1976) Preprint, U. of Heidelberg.
WALSH, J. B., The local time of the Brownian sheet, Asterisque v.52–53 (1978), p. 47–61.
For some applications to statistics
KIEFER, J., Skorokhod embedding of multivariate random variables and the sample distribution function, ZW 24 (1972) p. 1–35.
For interesting pictures of the Brownian sheet (and related processes)
MANDELBROJT, Fractals: from, chance and dimension, W.H. Freeman, San Francisco, 1977.
References
CAIROLI and WALSH: Stochastic integrals in the plane, Acta Math 134 (1975), 111–183.
GUYON and PRUM: Semi martingales a indice dans R2. Thesis, Université de Paris-sud, Orsay, 1980.
WONG and ZAKAI: Martingales and stochastic integrals for processes with a multidimensional parameter, ZW 29 (1974) 109–122.
_____: Weak martingales and stochastic integrals in the plane, Ann. Prob. 4 (1976) 570–587.
_____: An extension of stochastic integrals in the plane, Ann. Prob. 5 (1977) 770–778.
References For martingale representations
R. CAIROLI and J.B. WALSH, Stochastic integrals in the plane, Acta Math. 134, (1975), 111–183.
_____, Martingale representations and holomorphic processes, Ann. Prob. 5 (1977), 511–521.
K. ITO, Lectures on stochastic processes, Tata Institute.
E. WONG and M. ZAKAI, Martingales and stochastic integrals for processes with a multidimensional parameter, ZW. 29 (1974), 109–132.
M. YOR, Representation des martingales de carré intégrable relatives aux processes de Wiener et de Poisson, Z. W 35 (1976), 121–129.
For path continuity: In addition to the Acta article above, see
D. AKRY, Sur la regularité des trajectoires des martingales a deux indices, Z,W. 50 (1979), 149–157.
A. MILLET and L. SUCHESTON, Convergence et regularité des martingales multiples, C.R. 291 (1980), 147–150.
J. WALSH, Convergence and regularity of multiparameter martingales, ZW 46 (1979), 177–192.
E. WONG and M. ZAKAI, The sample function continuity of stochastic integrals in the plane, Ann. Prob. (1977), 1024–1027.
For stopping lines
R. CAIROLI and J.B. WALSH: Regions d'arret, localisations, et prolongements de martingales, ZW 44 (1978), 279–306.
E. WONG and M. ZAKAI: weak martingales and stochastic integrals in the plane, Ann Prob. 4 (1976), 570–586.
References
CAIROLI and WALSH: Stochastic integrals in the plane, Acta Math. 134 (1975), p. 111–183.
_____: Martingale representations and holomorphic processes, Ann. Prob. 5 (1977), 511–521.
GUYON and PRUM: Variations-produit et formule de Ito pour les-martingales representables a deux indices, Z.W. 56 (1981), 361–398.
WONG and ZAKAI, Differentation formulas for stochastic integrals in the plane, Stoch. Proc. and Appl. 16 (1978), 339–349.
References For holmorphic processes
CAIROLI and WALSH: Stochastic integrals in the plane, Acta Math. 134 (1975), 111–183.
_____: Martingale representations and holomorphic processes, Ann. Prob. 5(1977), 511–521.
_____: Prolongements de processus holomorphes, Séminaire de Strasbourg XI, Lecture Notes in Math 581, 340–348.
_____: Some examples of holomorphic processes. Seminaire de Strasbourg XI, Lecture Notes in Math 581, 327–339.
YOR: Etude de certains processus stochastiquement différentiables ou holomorphes, Ann. Inst. H. Poincaré 13 (1977), 1–25.
For related subjects
CAIROLI and WALSH: Regions d'arrêt, localisations et prolongements de martingales ZW 44 (1978), 279–306.
YOR: Formule de Cauchy relative à certains lacets browniens, Bull. Soc. Math. de France 105 (1977), 3–31.
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Walsh, J.B. (1986). Martingales with a multidimensional parameter and stochastic integrals in the plane. In: del Pino, G., Rebolledo, R. (eds) Lectures in Probability and Statistics. Lecture Notes in Mathematics, vol 1215. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0075875
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